arXiv:2607. 09235v1 Announce Type: cross Abstract: The current state of the art in AI/ML rests on deep neural architectures, which, in general, suffer from a lack of interpretability.
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The paper introduces SW-KAN, a Kolmogorov‑Arnold Network that replaces traditional B‑spline activations with Stieltjes‑Wigert q‑orthogonal polynomials defined on the semi‑infinite domain (0, ∞). It addresses the domain mismatch between unbounded inputs and bounded polynomial bases by applying a smooth exponential‑of‑tanh mapping, and uses a numerically stable three‑term recurrence to evaluate polynomial expansions efficiently. Experiments on image classification and continuous function approximation show that SW‑KAN achieves better accuracy‑efficiency trade‑offs than existing polynomial KANs, especially in resource‑constrained scenarios with limited data or feature dimensionality.
Polynomial-Augmented Neural Networks (PANNs) merge deep neural networks with polynomial expansions to leverage the flexibility of DNNs and the rapid convergence of polynomials. The architecture introduces orthogonality constraints, basis pruning, and polynomial preconditioning to stabilize training and improve accuracy across diverse problems. Experiments show that PANNs outperform both pure DNNs and polynomial methods in approximating smooth and limited‑smoothness functions, as well as in solving partial differential equations.
The paper discusses how backpropagation enables deep learning but does not inherently organize parameters for reusable functional components, leading to weight entanglement where overlapping parameter sets hinder independent modification. It introduces weight operators—parameterized modules that can be composed at inference—to address this, proposing a two-stage learning process that first infers operator composition and then updates only the selected operators. Vector Networks (VNs) are presented as an implementation that couples operator selection to local error-driven updates, demonstrating that learned operators can be recombined in unseen ways while keeping updates confined to the relevant parameter sets.
arXiv:2609.35938v1 Announce Type: new Abstract: This paper proposes an interpretable neural operator framework, the Kernel Operator Network (KernelOnet), which incorporates kernel functions explicitl...
NeuralCert presents a framework that learns high‑dimensional variational trial functions in a compact separable form, then spectrally diagnoses, prunes, and exactly certifies them via multimodular evaluation. The method is fully explicit and independently verifiable, and can run on a standard personal computer. Applied to three extremal problems, it demonstrates that neural optimization can discover better constructions, reveal empirical invariants useful for proofs, and expose optimization barriers that inspire new analytic or numerical approaches.
arXiv:2606. 26705v1 Announce Type: cross Abstract: Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes.